Such an instability has been observed both experimentally and numerically in a variety of rotating flows driven by mechanical forcings [1]}, [2]}, but never reported before in rotating convection systems. Motivated by geophysical and astrophysical applications, rotating convection has been studied extensively both for planar models [3]} and for spherical models [4]}, [5]}, [6]}, [7]}, [8]},
yet the nonlinear dynamics and turbulent regime remain to be fully understood [9]}. Recent numerical simulations [10]} and laboratory experiments [11]} of rotating convection in a cylinder using liquid gallium (\(\approx 0.025\) ) observed multi-modal interactions, yet the underlying mechanism remains to be elucidated.  [12]} numerically studied rotating convection in a sphere with \(Pr=0.0023\)  at moderate Ekman numbers and found that the non-linear interaction of various thermal inertial waves leads a weakly turbulent state, though no evidence of triadic resonances was shown. Here we show, through numerical simulations  at lower Ekman numbers, that the primary convective mode at low \(\)  (relevant to liquid metals) can excite a pair of inertial modes through the triadic resonance in a rotating sphere, leading to more complex convective motions. Increasing the thermal forcing leads to a turbulent state which shows signatures of inertial waves. Our study may offer a new pathway to study rotating turbulent convection within the framework of inertial wave turbulence [13]}, [14]}.
